Theorem: Frozen finite certificate

The complete evaluated contact image is

and

The original minor on rows two and three and columns ${\ell }_a,{\ell }_b$ is $64$.

If ${\alpha }_0$ has only

nonzero, then

Moreover,

and

Proof. All columns have zero first entry, so the rank is at most two. The two original columns of Lemma lem:pivots have last-two-row minor $\det \mathrm{diag}(8,8)=64$, giving rank at least two over both fields and spanning the asserted plane.

Their combination with coefficients $2/3,2/3$ is $m_S$. Appending the target therefore does not increase rank. An annihilator $z=(z_0,z_{+},z_{-})$ must satisfy $8z_{+}=8z_{-}=0$; conversely every $(z_0,0,0)$ annihilates the matrix. The target has the same blind first entry.

Subtracting ${\alpha }_0$ identifies the full affine fit family. Rank-nullity gives $15120-2=15118$.